sigma field generated by two random variables

Question

How do I show that the sigma field generated by two random variables, $(X,Y)$, same as sigma field generated by $(X+Y, Y)$?

Answer 1

Following this question, the sigma field generated by $X$ and $Y$ is the minimal sigma field containing sets of the form $\{X

Let $x,y\in \mathbb{R}$ be some arbitrary numbers. Sets $\{X

Similarly, for arbitrary $z,y\in \mathbb{R}$, we can write $\{X+Y < z\}=\{X

Therefore, those sigma fields are equal as each one is a subset of the other.